Optimal. Leaf size=158 \[ \frac {\text {Ci}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {Ci}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {Ci}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2} \]
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Rubi [A] time = 0.34, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4634, 4720, 4632, 3302} \[ \frac {\text {CosIntegral}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4632
Rule 4634
Rule 4720
Rubi steps
\begin {align*} \int \frac {x^4}{\cos ^{-1}(a x)^4} \, dx &=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {4 \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx}{3 a}+\frac {1}{3} (5 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2}-\frac {25}{6} \int \frac {x^4}{\cos ^{-1}(a x)^2} \, dx+\frac {2 \int \frac {x^2}{\cos ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \left (-\frac {\cos (x)}{4 x}-\frac {3 \cos (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}-\frac {25 \operatorname {Subst}\left (\int \left (-\frac {\cos (x)}{8 x}-\frac {9 \cos (3 x)}{16 x}-\frac {5 \cos (5 x)}{16 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^5}+\frac {125 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{96 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}+\frac {75 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac {2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac {5 x^5}{6 \cos ^{-1}(a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {\text {Ci}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac {27 \text {Ci}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac {125 \text {Ci}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 159, normalized size = 1.01 \[ \frac {80 a^5 x^5 \cos ^{-1}(a x)-64 a^3 x^3 \cos ^{-1}(a x)+192 a^2 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2+32 a^4 x^4 \sqrt {1-a^2 x^2}-400 a^4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2+2 \cos ^{-1}(a x)^3 \text {Ci}\left (\cos ^{-1}(a x)\right )+81 \cos ^{-1}(a x)^3 \text {Ci}\left (3 \cos ^{-1}(a x)\right )+125 \cos ^{-1}(a x)^3 \text {Ci}\left (5 \cos ^{-1}(a x)\right )}{96 a^5 \cos ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\arccos \left (a x\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 138, normalized size = 0.87 \[ \frac {5 \, x^{5}}{6 \, \arccos \left (a x\right )^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{3 \, a \arccos \left (a x\right )^{3}} - \frac {2 \, x^{3}}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{3} \arccos \left (a x\right )} + \frac {125 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{96 \, a^{5}} + \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{48 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 171, normalized size = 1.08 \[ \frac {\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )^{3}}+\frac {a x}{48 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arccos \left (a x \right )}+\frac {\Ci \left (\arccos \left (a x \right )\right )}{48}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{3}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \Ci \left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{48 \arccos \left (a x \right )^{3}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )^{2}}-\frac {25 \sin \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )}+\frac {125 \Ci \left (5 \arccos \left (a x \right )\right )}{96}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{3} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3} \int \frac {{\left (125 \, a^{4} x^{5} - 136 \, a^{2} x^{3} + 24 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{{\left (a^{5} x^{2} - a^{3}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + {\left (2 \, a^{2} x^{4} - {\left (25 \, a^{2} x^{4} - 12 \, x^{2}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + {\left (5 \, a^{3} x^{5} - 4 \, a x^{3}\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{6 \, a^{3} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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